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Numerical solution of Plateau’s problem of minimal surface using non-variational finite element method. The paper aims to discuss this issue.

An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.

The solutions obtained here are examined for different cases of non-linearity and are found sufficiently accurate.

The manuscript provide the non-variational solution for Plateau’s problem. Thus it has a good value in engineering application.

The purpose of this paper is to examine quadratic convergence of finite element analysis for hyperelastic material at finite strains via Abaqus‐UMAT as well as classification of the rates of convergence for iterative solutions in regular cases.

Different formulations for stiffness – Hessian form of the free energy functionals – are systematically given for getting the rate‐independent analytical tangent and the numerical tangent as well as rate‐dependent tangents using the objective Jaumann rate of Kirchoff stress tensor as used in Abaqus. The convergence rates for available element types in Abaqus are computed and compared for simple but significant nonlinear elastic problems, such as using the 8‐node linear brick (B‐bar) element – also with hybrid pressure formulation and with incompatible modes – further the 20‐node quadratic brick element with corresponding modifications as well as the 6‐node linear triangular prism element and 4‐node linear tetrahedral element with modifications.

By using the Jaumann rate of Kirchoff stress tensor for both, rate dependent and rate independent problems, quadratic or nearly quadratic convergence is achieved for most of the used elements using Abaqus‐UMAT interface. But in case of using rate independent analytical tangent for rate independent problems, even convergence at all is not assured for all elements and the considered problems.

First time the convergence properties of 3D finite elements available in Abaqus sre systematically treated for elastic material at finite strain via Abaqus‐UMAT.

This paper presents a numerical study of inverse parameter identification problems in fracture mechanics. Inverse methodology is applied to the detection of subsurface cracks and to the study of propagating cracks. The procedure for detecting subsurface cracks combines the finite element method with a sequential quadratic programming algorithm to solve for the unknown geometric parameters associated with the internal flaw. The procedure utilizes finite element substructuring capabilities in order to minimize the processing and solution time for practical problems. The finite element method and non‐linear optimization are also used in determining the direction a crack will propagate in a heterogeneous planar domain. This procedure involves determining the direction that produces the maximum strain energy release for a given increment of crack growth. The procedure is applied to several numerical examples. The results of these numerical studies coincide with theoretical predictions and experimentally observed crack behavior.

To reduce the computational complexity per step from O(n²) to O(n) for optimization based on quadratic surrogates, where n is the number of design variables.

Applying nonlinear optimization strategies directly to complex multidisciplinary systems can be prohibitively expensive when the complexity of the simulation codes is large. Increasingly, response surface approximations (RSAs), and specifically quadratic approximations, are being integrated with nonlinear optimizers in order to reduce the CPU time required for the optimization of complex multidisciplinary systems. For evaluation by the optimizer, RSAs provide a computationally inexpensive lower fidelity representation of the system performance. The curse of dimensionality is a major drawback in the implementation of these approximations as the amount of required data grows quadratically with the number n of design variables in the problem. In this paper a novel technique to reduce the magnitude of the sampling from O(n²) to O(n) is presented.

The technique uses prior information to approximate the eigenvectors of the Hessian matrix of the RSA and only requires the eigenvalues to be computed by response surface techniques. The technique is implemented in a sequential approximate optimization algorithm and applied to engineering problems of variable size and characteristics. Results demonstrate that a reduction in the data required per step from O(n²) to O(n) points can be accomplished without significantly compromising the performance of the optimization algorithm.

A reduction in the time (number of system analyses) required per step from O(n²) to O(n) is significant, even more so as n increases. The novelty lies in how only O(n) system analyses can be used to approximate a Hessian matrix whose estimation normally requires O(n²) system analyses.

This work aims to describe the physical and numerical modeling of a CFD solver for hypersonic flows in thermo-chemical non-equilibrium. This paper is the second of a two-part series that concerns the application of the solver introduced in Part I to adaptive unstructured meshes.

The governing equations are discretized with an edge-based stabilized finite element method (FEM). Chemical non-equilibrium is simulated using a laminar finite-rate kinetics, while a two-temperature model is used to account for thermodynamic non-equilibrium. The equations for total quantities, species and vibrational-electronic energy conservation are loosely coupled to provide flexibility and ease of implementation. To accurately perform simulations on unstructured meshes, the non-equilibrium flow solver is coupled with an edge-based anisotropic mesh optimizer driven by the solution Hessian to carry out mesh refinement, coarsening, edge swapping and node movement.

The paper shows, through comparisons with experimental and other numerical results, how FEM + anisotropic mesh optimization are the natural choice to accurately simulate hypersonic non-equilibrium flows on unstructured meshes. Three-dimensional test cases demonstrate how, for high-speed flows, shocks resolution, and not necessarily boundary layers resolution, is the main driver of solution accuracy at walls. Equally distributing the error among all elements in a suitably defined Riemannian space yields highly anisotropic grids that feature well-resolved shock waves. The resulting high level of accuracy in the computation of the enthalpy jump translates into accurate wall heat flux predictions. At the opposite end, in all cases examined, high-quality but isotropic unstructured meshes gave very poor solutions with severely inadequate heat flux distributions not even featuring expected symmetries. The paper unequivocally demonstrates that unstructured anisotropically adapted meshes are the best, and may be the only, way for accurate and cost-effective hypersonic flow solutions.

Although many hypersonic flow solvers are developed for unstructured meshes, few numerical simulations on unstructured meshes are presented in the literature. This work demonstrates that the proposed approach can be used successfully for hypersonic flows on unstructured meshes.

In this paper the use of classical strain measures in analysis of trusses at finite deformations will be discussed. The results will be compared to the ones acquired using a novel strain measure based on the Hyperbolic Sine function. Through the evaluation of results, algebraic development and graph analysis, the properties of the Hyperbolic Sine strain measure will be examined.

Through graph plotting, comparisons between the novel strain measure and the classic ones will be made. The formulae for the implementation of the Hyperbolic Sine strain measure into a positional finite element method are developed. Four engineering applications are presented and comparisons between results obtained using all strain measures studied are made.

The proposed strain measure, Hyperbolic Sine, has objectivity and symmetry. The linear constitutive model formed by the Hyperbolic Sine strain and its conjugated stress presents an increasing stiffness, both in compression and tension, a behavior that can be useful in the modeling of several materials.

The structural analysis performed on the four examples of trusses in this article did not consider the variation of the cross-sectional area of the elements or the buckling phenomenon, moreover, only elastic behavior is considered.

The present article proposes the use of a novel strain measure family, based on the Hyperbolic Sine function and suitable for structural applications. Mathematical expressions for the use of the Hyperbolic Sine strain measure are established following the energetic concepts of the positional formulation of the finite element method.

Gives introductory remarks about chapter 1 of this group of 31 papers, from ISEF 1999 Proceedings, in the methodologies for field analysis, in the electromagnetic community. Observes that computer package implementation theory contributes to clarification. Discusses the areas covered by some of the papers ‐ such as artificial intelligence using fuzzy logic. Includes applications such as permanent magnets and looks at eddy current problems. States the finite element method is currently the most popular method used for field computation. Closes by pointing out the amalgam of topics.

This is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. The authors trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods.

Starting from the global optimum of the compliance minimization problem, the authors employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum and how the initial guess has some weight in determining the final optimum.

The non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application.

In this article, the authors present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. The authors put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.

This paper aims to present a nonlinear finite element model (FEM) of the Brushless DC (BLDC) motor and the application of the optimal linear–quadratic control-based method to determine the excitation voltage and current waveform considering the minimization of the energy injected to the input circuit and energy lost. The control problem is designed and analyzed using the feedback gain strategy for the infinite time horizon problem.

The method exploits the distributed parameters, nonlinear FEM of the device. First, dynamic equations of the BLDC motor are transformed into a suitable form that makes an ARE (algebraic Riccati equation)-based control technique applicable. Moreover, in the controller design, a Bryson scaling method is used to obtain desirable properties of the closed-loop system. The numerical techniques for solving ARE with the gradient damping factor are proposed and described. Results for applied control strategy are obtained by simulations and compared with measurement.

The proposed control technique can ensure optimal dynamic response, small steady-state error and energy saving. The effectiveness of the proposed control strategy is verified via numerical simulation and experiment.

The authors introduced an innovative approach to the well-known control methodology and settled their research in the newest literature coverage for this issue.

This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed.

A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation.

The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model.

A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.